| 635 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 636 |
|
a \emph{symplectic} flow if it satisfies, |
| 637 |
|
\begin{equation} |
| 638 |
< |
'\varphi^T J '\varphi = J. |
| 638 |
> |
{\varphi '}^T J \varphi ' = J. |
| 639 |
|
\end{equation} |
| 640 |
|
According to Liouville's theorem, the symplectic volume is invariant |
| 641 |
|
under a Hamiltonian flow, which is the basis for classical |
| 643 |
|
field on a symplectic manifold can be shown to be a |
| 644 |
|
symplectomorphism. As to the Poisson system, |
| 645 |
|
\begin{equation} |
| 646 |
< |
'\varphi ^T J '\varphi = J \circ \varphi |
| 646 |
> |
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 647 |
|
\end{equation} |
| 648 |
|
is the property must be preserved by the integrator. |
| 649 |
|
|
| 685 |
|
ordinary implicit Runge-Kutta methods are not suitable for |
| 686 |
|
Hamiltonian system. Recently, various high-order explicit |
| 687 |
|
Runge--Kutta methods have been developed to overcome this |
| 688 |
< |
instability \cite{}. However, due to computational penalty involved |
| 689 |
< |
in implementing the Runge-Kutta methods, they do not attract too |
| 690 |
< |
much attention from Molecular Dynamics community. Instead, splitting |
| 691 |
< |
have been widely accepted since they exploit natural decompositions |
| 692 |
< |
of the system\cite{Tuckerman92}. |
| 688 |
> |
instability. However, due to computational penalty involved in |
| 689 |
> |
implementing the Runge-Kutta methods, they do not attract too much |
| 690 |
> |
attention from Molecular Dynamics community. Instead, splitting have |
| 691 |
> |
been widely accepted since they exploit natural decompositions of |
| 692 |
> |
the system\cite{Tuckerman92}. |
| 693 |
|
|
| 694 |
|
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
| 695 |
|
|
| 744 |
|
symmetric property, |
| 745 |
|
\begin{equation} |
| 746 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
| 747 |
< |
\lable{introEquation:timeReversible} |
| 747 |
> |
\label{introEquation:timeReversible} |
| 748 |
|
\end{equation} |
| 749 |
|
|
| 750 |
|
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
| 802 |
|
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
| 803 |
|
\label{introEquation:positionVerlet1} \\% |
| 804 |
|
% |
| 805 |
< |
q(\Delta t) = q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
| 805 |
> |
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
| 806 |
|
q(\Delta t)} \right]. % |
| 807 |
|
\label{introEquation:positionVerlet1} |
| 808 |
|
\end{align} |
| 828 |
|
\] |
| 829 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 830 |
|
can obtain |
| 831 |
< |
\begin{eqnarray} |
| 831 |
> |
\begin{eqnarray*} |
| 832 |
|
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
| 833 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 + |
| 834 |
< |
h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 + \ldots ) |
| 835 |
< |
\end{eqnarray} |
| 833 |
> |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 834 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
| 835 |
> |
\ldots ) |
| 836 |
> |
\end{eqnarray*} |
| 837 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 838 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 839 |
|
procedure can be applied to general splitting, of the form |
